Cusp excursions on parameter spaces (Q2840164)

Interactions between the ergodic theory of flows on homogeneous spaces of Lie groups and Diophantine approximation form a thriving area of current research. Following works of Dani and Kleinbock-Margulis, there is now a robust and very useful dictionary called the ``Dani correspondence'' which relates Diophantine properties of vectors in \(\mathbb{R}^n\) and cuspidal excursions of orbits of certain partially hyperbolic flows on \(\text{SL}_{n+1}(\mathbb R)/\text{SL}_{n+1}(\mathbb Z)\). This well-written paper provides an axiomatic and elementary treatment of some important instances of this connection, especially in the setting of ``horospherical flows''. The author proves interesting results relating Diophantine exponents and dynamical properties of associated orbits. He also proves a logarithm law for horocycle flows on moduli spaces of quadratic differentials and obtains bounds for the deviation of ergodic averages for horocycle flows.